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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Witsenhausen counterexample ： ウィキペディア英語版 Witsenhausen's counterexample Witsenhausen's counterexample, shown in the figure below, is a deceptively simple toy problem in decentralized stochastic control. It was formulated by Hans Witsenhausen in 1968.〔Witsenhausen, Hans. "A counterexample in stochastic optimum control." ''SIAM J. Control'', Volume 6, Issue 1, pp. 131–147 (February 1968)〕 It is a counterexample to a natural conjecture that one can generalize a key result of centralized linear-quadratic-Gaussian control systems: that affine (linear) control laws are optimal. Witsenhausen showed that there exist nonlinear control laws that outperform all linear laws in a decentralized context. The problem of finding the optimal control law remains unsolved.〔Ho, Yu-Chi, "Review of the Witsenhausen problem". ''Proceedings of the 47th IEEE Conference on Decision and Control (CDC)'', pp. 1611–1613, 2008.〕 File:WitsenhausenCounterexample.jpg ==Statement of the counterexample== The statement of the counterexample is simple: two controllers attempt to control the system by attempting to bring the state close to zero in exactly two time steps. The first controller observes the initial state $x\_0.$ There is a cost on the input $u\_1$ of the first controller, and a cost on the state $x\_2$ after the input of the second controller. The input $u\_2$ of the second controller is free, but it is based on noisy observations $y\_1=x\_1+z$ of the state $x\_1$ after the first controller's input. The second controller cannot communicate with the first controller and thus cannot observe either the original state $x\_0$ or the input $u\_1$ of the first controller. Thus the system dynamics are :$x\_1=x\_0+u\_1,$ :$x\_2=x\_1-u\_2,$ with the second controller's observation equation :$y\_1=x\_1+z.$ The objective is to minimize an expected cost function, :$k^2E()+E()$, where the expectation is taken over the randomness in the initial state $x\_0$ and the observation noise $z$, which are distributed independently. The observation noise $z$ is assumed to be distributed in a Gaussian manner, while the distribution of the initial state value $x\_0$ differs depending on the particular version of the problem. The problem is to find control functions :$u\_1(x\_0)\; \backslash quad\; \backslash text\; \backslash quad\; u\_2(y\_1)$ that give at least as good a value of the objective function as do any other pair of control functions. Witsenhausen showed that the optimal functions $u\_1(x\_0)$ and $u\_2(y\_1)$ cannot be linear. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Witsenhausen's counterexample」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.